Real Numbers and Convergence to Non-Real Complex Numbers

It is often asked whether a series of real numbers can converge to a non-real complex number, such as i. In this article, we explore the nature of convergence in the complex plane and prove that it is impossible for a series of real numbers to converge to a non-real complex number, specifically isqrt{-1}.

Understanding the Complex Plane

The complex plane is a fundamental concept in complex analysis, where the real numbers form a straight line, known as the real axis. The imaginary unit i is located on a circle of radius 1, specifically at the top of the unit circle. This point is precisely 1 unit away from the real axis.

No Real Sequence Converges to i

Geometric Interpretation in the Complex Plane

Imagine drawing a neighborhood of radius 0.5 around the point i in the complex plane. This neighborhood will capture a small circle centered at i with a radius of 0.5. Notably, there are no real numbers within this circle since the real axis is the real number line, and the point i is not on this line. Thus, a sequence of real numbers, by its nature, cannot converge to i.

Proof of Non-Convergence

Let's consider a sequence of real numbers a_j. Suppose this sequence converges to a non-real complex number, specifically i. We will show a contradiction by setting a small distance ε 0.1.

By the definition of convergence, there exists an N such that for all j N, the distance between a_j and i is less than 0.1:

left|a_j - i right|

Squaring both sides and simplifying, we get:

|a_j^2 - 1|

Since a_j is a real number, its square a_j^2 is also real. The distance from a_j^2 to 1 must be less than 0.01:

|a_j^2 - 1|

This implies that a_j^2 is very close to 1. However, for any real number a_j, the absolute value of a_j^2 must be at least 1:

|a_j^2| 1

Therefore, the distance between a_j^2 and 1 cannot be less than 0.01 while still satisfying the condition |a_j^2| 1. This is a contradiction.

Conclusion

In summary, a series of real numbers can only converge to a real number. A non-real complex number, such as i, is separated from the real axis by a positive distance that cannot be overcome by any convergent sequence of real numbers. Mathematically, this distance is precisely 1 unit, as i is on the unit circle in the complex plane.